These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Label |
Polynomial $/ \Q_p$ |
Galois group $/ \Q_p$ |
Galois degree $/ \Q_p$ |
$\#\Aut(K/\Q_p)$ |
Artin slope content $/ \Q_p$ |
Swan slope content $/ \Q_p$ |
Hidden Artin slopes $/ \Q_p$ |
Hidden Swan slopes $/ \Q_p$ |
Ind. of Insep. $/ \Q_p$ |
Assoc. Inertia $/ \Q_p$ |
Resid. Poly |
Jump Set |
| 2.2.6.22a1.6 |
$( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{3} + 8 x + 2$ |
$(C_6\times C_2):C_2$ (as 12T15) |
$24$ |
$6$ |
$[2, 3]_{3}^{2}$ |
$[1,2]_{3}^{2}$ |
$[2]$ |
$[1]$ |
$[6, 0]$ |
$[1, 1]$ |
$z^4 + z^2 + 1,z + 1$ |
$[3, 9]$ |
| 2.2.6.22a2.38 |
$( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x + 2$ |
$C_6\wr C_2$ (as 12T42) |
$72$ |
$6$ |
$[2, 3]_{3}^{6}$ |
$[1,2]_{3}^{6}$ |
$[2]^{3}$ |
$[1]^{3}$ |
$[6, 0]$ |
$[1, 1]$ |
$z^4 + z^2 + 1,z + t$ |
$[3, 9]$ |
| 2.2.6.22a2.42 |
$( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 6 ( x^{2} + x + 1 ) + 2 x + 2$ |
$C_6\wr C_2$ (as 12T42) |
$72$ |
$6$ |
$[2, 3]_{3}^{6}$ |
$[1,2]_{3}^{6}$ |
$[2]^{3}$ |
$[1]^{3}$ |
$[6, 0]$ |
$[1, 1]$ |
$z^4 + z^2 + 1,z + t$ |
$[3, 9]$ |
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